[Paper Review] 2 Realization theory of discrete-time linearswitched systems

M. Petreczky|arXiv (Cornell University)|Aug 18, 2016

Control Systems and Identification54 references18 citations

TL;DR

This paper establishes realization theory for discrete-time linear switched systems (DTLSSs), proving that an input-output map admits a DTLSS realization if and only if its Hankel matrix has finite rank. The key contribution is a characterization of minimality via span-reachability and observability, with minimal realizations uniquely isomorphic up to state-space transformation, using rational formal power series as the core mathematical tool.

ABSTRACT

The paper presents realization theory of discrete-time linear switched systems. A discrete-time linear switched system is a hybrid system, such that the continuous sub-system associated with each discrete state is linear. In this paper we present necessary and sufficient conditions for an input-output map to admit a discrete-time linear switched state-space realization. The conditions are formulated as finite rank conditions of a generalized Hankel-matrix. In addition, we present a characterization of minimality of discrete-time linear switched systems in terms of reachability and observable.Further, we prove that minimal realizations are unique up to isomorphism. We also discuss procedures for converting a linear switched system to a minimal one and we present an algorithm for constructing a state-space representation from input-output data.The paper uses the theory rational formal power series in non-commutative variables. The latter theory was successfully applied to bilinear and state-affine systems in the past.

Motivation & Objective

• Address the lack of realization theory for discrete-time linear switched systems, despite extensive work on continuous-time cases.
• Provide necessary and sufficient conditions for an input-output map to admit a DTLSS state-space realization.
• Characterize minimality of DTLSSs in terms of span-reachability and observability, and prove uniqueness of minimal realizations up to isomorphism.
• Develop algorithms for minimizing DTLSSs and constructing state-space models from input-output data.
• Establish a formal correspondence between DTLSSs and rational formal power series, enabling the use of advanced algebraic tools.

Proposed method

• Define DTLSSs as discrete-time systems with linear dynamics per mode, where switching is governed by an external input.
• Introduce the concept of Markov-parameters and Hankel matrix for DTLSSs, generalizing linear system theory to non-commutative settings.
• Use the theory of rational formal power series in non-commutative variables as the foundational mathematical framework.
• Prove that a DTLSS realization exists if and only if the Hankel matrix of the input-output map has finite rank.
• Construct a minimization algorithm that transforms any DTLSS into a minimal one via sequential reachability and observability reduction.
• Establish isomorphism between DTLSSs and their associated representations of rational formal power series, enabling structural equivalence.

Experimental results

Research questions

• RQ1 Under what conditions does an input-output map admit a discrete-time linear switched state-space realization?
• RQ2 How can minimality of a DTLSS be characterized in terms of system-theoretic properties like reachability and observability?
• RQ3 Are minimal realizations of a given input-output map unique up to isomorphism?
• RQ4 What is the relationship between the realization problem for DTLSSs and the theory of rational formal power series?
• RQ5 How can one algorithmically construct a minimal DTLSS from input-output data?

Key findings

• An input-output map admits a DTLSS realization if and only if its associated Hankel matrix has finite rank.
• Minimal DTLSSs are characterized by both span-reachability and observability, and such realizations are unique up to isomorphism.
• Minimal realizations are isomorphic to the minimal representations of the corresponding rational formal power series.
• A minimization algorithm exists that transforms any DTLSS into a minimal one while preserving its input-output behavior.
• A procedure is provided to construct a state-space representation directly from the Hankel matrix of the input-output map.
• The realization problem for DTLSSs is equivalent to the realization problem for rational formal power series, establishing a deep algebraic correspondence.